I. Introduction: A historical view

One can hardly discuss the impact of chaos on mathematics and
ignore the impact of mathematics on chaos. Looking at the problem more
carefully, one realizes that there is indeed a rich history of interactions
between the mathematical theory of dynamical systems, and its applications in
the field of chaos. Such a situation prevails rather generally for the relations
of mathematics with its applications, but it is particularly striking here
because chaos is, basically, a mathematical phenomenon. It will therefore be
worth discussing briefly the history of the complex relations between the
mathematics of dynamical systems theory, and the analysis of chaos in physics
and other sciences.

We start with some definitions. A dynamical system is simply a
time evolution defined by a differential equation

(1)

in the case of a continuous time, or by an equation of the form

x

t+1=f(xt)
(2)

in the case of a discrete time t. The variable x belongs to
some finite or infinite dimensional space M. Notice that the time does not
explicitly occur in the right-hand sides of (1) and (2): we consider autonomous
systems. There is thus a (nonlinear) time evolution operator
ft such that
x(t)=ftx(0). In the discrete time
case, ft is simply the tth iterate of f.
We have

f

0=identity,
ft·fs=ft+s

We shall assume that F in (1) or f in (2) is differentiable. This
looks like a modest technical assumption, but it is in fact the key to
interesting developments.

Consider the change dx(t) corresponding to a
change dx(0) in initial condition. We have

Typically - and we shall not be more precise at this point - both
dx(t) and ¶f/¶x behave
exponentially with time, i.e.

dx(t) ~
elt.

When l<0, we have stability, when l>0 we have
sensitive dependence on initial condition. It is clear how this can happen at an
unstable fixed point, for instance, but what is more surprising is that
sensitive dependence on initial condition can occur for (almost) all initial
condition. This prevalence of sensitivity is what we now call chaos.

Chaos, as we have just defined it, was discovered by J. Hadamard
at the end of the nineteenth century in a special (Hamiltonian) dynamical system
called the geodesic flow on a manifold of negative curvature [1]. Hadamard
immediately understood the philosophical importance of his result: an
arbitrarily small uncertainty on the initial condition entails a large
uncertainty on the predicted state of the system after a sufficiently long time.
Other scientists (P. Duhem, H. Poincaré) also understood the importance of the
phenomenon discovered by Hadamard, and Poincaré [2] discusses the relevance of
sensitive dependence on initial condition to the dynamics of a hard sphere
system, and to weather predictions.

The early discovery of chaos had however no lasting influence on
physics. The new ideas were forgotten and had to be rediscovered again, much
later and independently. On the mathematical side, however, the work of Hadamard
and Poincaré led to uninterrupted progress up to the present day, with
contributions of such people as Kolmogorov, Smale, and many more. Incidentally,
an essential step in the mathematical development of dynamical systems theory
was the creation of ergodic theory, for which ideas originating in
physics were important.

The time evolution of chaotic systems is typically complicated and
irregular-looking. Indeed, a regular time evolution is predictable and therefore
not chaotic. When the interest for complicated and irregular time evolutions
developed among physicists in the 1970s, to give what is now called the theory
of chaos, all kinds of new scientific tools existed that had not been available
to Poincaré. One such tool is the electronic computer, which allowed Lorenz [3]
to compute in 1963 a chaotic time evolution, and visualize it in the form of
what we now call a strange attractor. Other tools were mathematical, like
ergodic theory. Finally, there were new experimental tools permitting for
instance a detailed study of the onset of hydrodynamic turbulence. It is this
experimental study that showed that hydrodynamical tubulence is chaotic, as we
would now say, corresponding to the claim of Ruelle and Takens [4], in 1971,
that it is described by strange attractors.

It must be said at this point that, however insightful and
brilliant, the physical ideas of Poincaré on chaos were at the level of
scientific philosophy. Because new tools are available, our present
understanding of chaos in physics is at the level of quantitative science. And
attempted applications of chaos to biology, economics, etc., which are now at
the level of scientific philosophy, may later reach the level of quantitative
science.

I have stressed up to now the idea that the study of chaos
consisted in applying available mathematics to the understanding of natural
phenomena. But there have also been important influences in the opposite
direction: from the physics of chaos to mathematics. New mathematical tools were
created because they were needed to understand chaos, or new ideas coming from
chaos were found to have surprisingly important mathematical content, and were
then developed for their own sake. In fact, the limit between physical and
mathematical ideas is often unclear and the same people have been seen sometimes
functioning as physicists, sometimes as mathematicians.

In what follows I shall give a list of some mathematical
developments where the ideas of chaos played a significant role. The list does
not attempt at being exhaustive, and the scope and nature of the different items
is rather
different.